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Finite projective spaces and planes
Further information on finite projective planes: Projective plane § Finite projective planes

For finite projective spaces of dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, GF(q), whose order
(that is, number of elements) is q (a prime power). A finite projective space defined over such a finite field has q + 1 points on a line, so the two concepts of order coincide. Notationally, PG(n, GF(q)) is usually written as PG(n, q).

All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field.

The smallest projective plane is the Fano plane, PG(2, 2) with 7 points and 7 lines. The smallest 3-dimensional projective spaces is PG(3,2), with 15 points, 35 lines and 15 planes.

Algebraic geometry
An important property of projective spaces and projective varieties is that the image of a projective variety under a morphism of algebraic varieties is closed for Zariski topology (that is, it is an algebraic set). This is a generalization to every ground field of the compactness of the real and complex projective space.

A projective space is itself a projective variety, being the set of zeros of the zero polynomial.

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